Wavelet Galerkin method in multi-scale homogenization of heterogeneous media

The hierarchical properties of scaling functions and wavelets can be utilized as effective means for multi-scale homogenization of heterogeneous materials under Galerkin framework. It is shown in this work, however, when the scaling functions are used as the shape functions in the multi-scale wavelet Galerkin approximation, the linear dependency in the scaling functions renders improper zero energy modes in the discrete differential operator (stiffness matrix) if integration by parts is invoked in the Galerkin weak form. An effort is made to obtain the analytical expression of the improper zero energy modes in the wavelet Galerkin differential operator, and the improper nullity of the discrete differential operator is then removed by an eigenvalue shifting approach. A unique property of multi-scale wavelet Galerkin approximation is that the discrete differential operator at any scale can be effectively obtained. This property is particularly useful in problems where the multi-scale solution cannot be obtained simply by a wavelet projection of the finest scale solution without utilizing the multi-scale discrete differential operator, for example, the multi-scale analysis of an eigenvalue problem with oscillating coefficients.